One remark that Penelope Maddy makes several times in Naturalism in Mathematics, is that if the indispensability argument was really important in justifying mathematics, then set theorists should be looking to debates over quantum gravity to settle questions of new axioms. Since this doesn’t seem to be happening, she infers that the indispensability argument can’t play the role Quine and Putnam (and perhaps her earlier book?) argued that it does.
However, since the awarding of the Fields Medal to the physicist Ed Witten in 1990, it’s not totally clear that Maddy is right about this. Set theorists certainly don’t pay much attention to string theory and related theories, but other mathematicians in low-dimensional topology and algebraic geometry seem to. I don’t know much about the details, but from what I understand, physicists have conjectured some deep and interesting connections between seemingly disparate areas of mathematics, in order to explain (or predict?) particular physical phenomena. These connections have rarely been rigorously proved, but they have stimulated mathematical research both in pursuing the analogies and attempting to prove them. Although the mathematicians often find the physicists’ work frustratingly imprecise and non-rigorous, once the analogies and connections have been suggested by physicists, mathematicians get very interested as well.
If hypothetically, one of these connections was to turn out to be independent of ZFC, I could imagine that there would at least be a certain camp among mathematicians that would take this as evidence for whatever large cardinal (or other) principle was needed to prove the connection. Set theorists themselves haven’t paid too much attention to these issues, because the interesting connections are in mathematical areas traditionally considered quite distant from set theory. Instead, they have traditionally looked at intra-set-theoretic considerations to justify large cardinals. But if it became plausible that some of these other debates would turn out to be connected, I’m sure they would start paying attention to the physics research, contrary to what Maddy suggests.